1. Field of the Invention
The present invention relates to a method for fast generation of a geostatistical reservoir model on a flexible grid, representative of a porous heterogeneous medium.
2. Description of the Prior Art
Optimization of the development of petroleum reservoirs requires a precise survey of the subsoil. It is therefore necessary to generate a reservoir model compatible, as much as possible, with all of the data collected (logs, seismic data, outcrop data, production records, . . . ) and to estimate, by reservoir simulation, the production of this constrained reservoir. The goal is to reduce the uncertainty on the production predictions insofar as a reservoir model compatible with all of the data is available. Since the data collected are insufficient for deterministic construction of a reservoir model, stochastic modelling techniques, most often based on geostatistical techniques, are used, which provide a family of numerical stochastic models of the reservoir. Prior to reservoir simulation, a reservoir characterization in terms of geometry and petrophysical properties is carried out in form of numerical stochastic reservoir models. A numerical model consists of an N-dimension grid (N>0 and generally equals two or three) each cell of which is assigned the value of a petrophysical property characteristic of the studied zone. It can be, for example, the porosity or the permeability distributed in a reservoir. What is referred to as the map of a petrophysical property is all of the values of the petrophysical property expressed in each one of the cells of the grid. Thus, sampling of a map follows gridding of the numerical model. A numerical reservoir model is the combination of a grid and of petrophysical property maps of the reservoir. In a stochastic context, the term realization of a petrophysical property will be used rather than the term map. There is an infinity of possible realizations compatible with one geologic description of the medium.
Two main families can be clearly distinguished from among all the data required for characterization of a reservoir: the data referred to as static, which include 3D seismic data, outcrop and log data, and the dynamic data which comprise production records and 4D seismic data. The dynamic data are distinguished in that they vary with time according to fluid flows. Integration of these two data families within the numerical model is not performed identically. Integration of the static data is carried out upon generation of the geologic model, whereas integration of the dynamic data is carried out via an inversion problem and requires many flow simulations.
The procedure for conditioning the numerical models to the production data requires frequent use of simulators. The numerical model therefore has to comprise few cells in order to obtain short calculating times. However, flow simulations on the numerical model have to be very precise in the neighborhood of singular objects such as faults, clay banks or wells for example. The grids therefore have to be very precise for these zones.
The method allowing generation of a numerical geostatistical model representative of a porous heterogeneous medium therefore has to meet several requirements:
1. The method must allow, if necessary, integration of static well data;
2. The method must allow, if necessary, integration of dynamic production data;
3. The method must limit errors on flow simulations in the neighborhood of singular objects.
1. Static Data Integration
It is sometimes necessary to constrain the simulations by static data such as well data. Conditioning by static data consists in constraining realization Y by values Vi at points Pi. There are many techniques for estimating locally the geostatistical parameters at points Pi. Kriging techniques such as simple kriging, ordinary kriging or universal kriging can be mentioned for example.
2. Dynamic Production Data Integration
An inverse problem has to be solved and a cost function therefore has to be minimized to integrate the dynamic production data in the model. Such a technique is for example described in the following document:
Tarantola, A., 1987, “Inverse Problem Theory—Methods for Data Fitting and Model Parameter Estimation”, Elsevier, Amsterdam, p. 613.
However, these techniques generally involve an objective function defined by a large number of parameters. On the other hand, a parameterization method allows fast integration of the dynamic data on the basis of a cost function described by a small number of parameters. It is the gradual deformation technique described, for example, in French patent 2,780,798 filed by the assignee.
It should be noted that this method is a parameterization technique allowing deformation of a realization from a reduced number of parameters while keeping the geostatistical structure thereof. Owing to these properties, this parameterization technique is of great interest for constraining a reservoir model by dynamic production data. It has in fact been introduced in the case history calibration process intended for construction of a reservoir model constrained by the production data. With this gradual deformation technique, it is possible to modify both globally and locally the geostatistical realizations. In the case of local modifications, knowledge of the underlying Gaussian white noise is necessary.
3. Limitation of the Error on Flow Simulations in the Neighborhood of Singular Objects
In order to limit the error of the results of flow simulations due to the grid, the petroleum industry turned to the use of flexible grids. A flexible grid is a grid containing cells of different shapes and different volumes. This type of grid is for example described in the following document:
Balaven-Clermidy, S., 2001, “Génération de Maillages Hybrides pour la Simulation des Réservoirs Pétroliers”, PhD thesis, Ecole des Mines de Paris.
This type of grid allows improvement of the description of the numerical model by fining down the flow model in the most flow-sensitive zones (wells, faults, high-heterogeneity zones, . . . ).
The local or global gradual deformation technique is thus clearly the most efficient technique for integrating dynamic data. Besides, in order to limit information loss upon creation of the grid used for flow simulations, it is necessary to use flexible grids. However, all geostatistical simulation methods are not fully compatible with the gradual deformation method. For example, realizations simulated from the turning band method cannot be subjected to a local gradual deformation because geostatistical turning band type simulators do not separate the Gaussian white noise generation from the covariance structure. As for the methods based on a Cholesky decomposition of the covariance matrix, computer processing cannot be considered for a number of points above 1000. Now, the size of a geostatistical model is generally of the order of one million grid cells. These methods can therefore not be considered for an oil reservoir survey. Sequential simulation algorithms can be adapted to the gradual deformation method although they require some adjustments concerning the definition of the path for visiting the points where a value is to be simulated. This technique is for example described in the following document:
Hu, L. Y., Blanc G. and Noetinger B., 2001; “Gradual Deformation and Iterative Calibration of Sequential Stochastic Simulations”, Math. Geol., 33(4).
However, the drawback of this method is that the simulations at the N-th point involve their conditioning by the previously simulated N−1 values: the algorithm can rapidly become costly even when using moving neighborhood techniques. Another drawback is that it implies cells of substantially equivalent size.
The FFT-MA method is naturally integrated in the context of gradual deformation. It is described in:
Le Ravalec, M., Noetinger, B. And Hu, L.-Y., 2000, “The FFT-Moving Average (FFT-MA)Generator: An Efficient Tool for Generating and Conditioning Gaussian Simulations”, Math. Geol., 32(6), 701-723.
This method produces realizations which can be deformed globally or locally according to the gradual deformation principles. This property is linked with the convolution product which uncouples the random numbers (that is the Gaussian white noise) from the geostatistical parameters (that is the core of the covariance function). The random numbers can then be varied either in their entirety, or by zone, and global or local deformations can be induced while keeping the covariance model. However, this technique cannot be used on flexible grids by construction. In fact, it is based on the use of fast Fourier transforms, which are particularly efficient tools in the case of Cartesian grids.